\chapter{Analysis Algorithms}
\label{ch:algorithms}

\section{Call Graph Construction}

The construction of function call graphs represents the core analytical capability of the system. This process transforms the parsed function definitions and call relationships into structured graph representations that enable comprehensive analysis of code organization and dependencies.

\subsection{Graph Theoretical Foundations}

The call graph construction process is grounded in graph theory, with specific adaptations for the characteristics of software call relationships:

\begin{definition}[Directed Call Graph]
A directed call graph $G = (V, E, W)$ is defined as:
\begin{itemize}
\item $V = \{f_1, f_2, ..., f_n\}$ represents the set of function vertices
\item $E \subseteq V \times V$ represents directed edges indicating call relationships
\item $W: E \rightarrow \mathbb{N}$ assigns weights to edges representing call frequency or significance
\end{itemize}
where each edge $(f_i, f_j) \in E$ indicates that function $f_i$ directly calls function $f_j$.
\end{definition}

\begin{definition}[Call Path]
A call path $P$ from function $f_s$ to function $f_t$ is a sequence of functions $P = \langle f_s, f_1, f_2, ..., f_k, f_t \rangle$ such that $(f_i, f_{i+1}) \in E$ for all consecutive pairs in the sequence.
\end{definition}

\subsection{Construction Algorithm}

The call graph construction algorithm processes parsed function and call data to build comprehensive graph structures:

\begin{algorithm}
\caption{Call Graph Construction}
\label{alg:call-graph-construction}
\begin{algorithmic}[1]
\Require Function set $F$, Call relationship set $C$, Configuration $\text{config}$
\Ensure Call graph $G = (V, E, W)$

\State $V \gets \emptyset$, $E \gets \emptyset$, $W \gets \{\}$
\State $\text{nameResolver} \gets \text{InitializeNameResolver}(F)$

\Comment{Phase 1: Vertex Creation}
\For{each function $f \in F$}
    \If{$\text{ShouldInclude}(f, \text{config})$}
        \State $v \gets \text{CreateVertex}(f)$
        \State $V \gets V \cup \{v\}$
    \EndIf
\EndFor

\Comment{Phase 2: Edge Creation}
\For{each call relationship $c \in C$}
    \State $\text{caller} \gets \text{ResolveFunction}(c.\text{caller}, \text{nameResolver})$
    \State $\text{callee} \gets \text{ResolveFunction}(c.\text{callee}, \text{nameResolver})$
    
    \If{$\text{caller} \in V \land \text{callee} \in V$}
        \State $e \gets (\text{caller}, \text{callee})$
        \State $E \gets E \cup \{e\}$
        \State $W[e] \gets W[e] + 1$ \Comment{Increment call frequency}
    \EndIf
\EndFor

\Comment{Phase 3: Graph Validation and Cleanup}
\State $G \gets \text{ValidateGraph}(V, E, W)$
\State $G \gets \text{ApplyFilters}(G, \text{config})$

\Return $G$
\end{algorithmic}
\end{algorithm}

\subsection{Name Resolution and Disambiguation}

One of the most challenging aspects of call graph construction involves resolving function names in the presence of overloading, namespaces, and template instantiations:

\paragraph{Namespace Resolution} The system maintains a namespace context stack during parsing to resolve qualified function names accurately. This includes handling using declarations, namespace aliases, and argument-dependent lookup (ADL).

\paragraph{Overload Resolution} When multiple functions share the same name, the system attempts to resolve calls to specific overloads based on parameter type analysis. This is particularly important for C++ codebases with extensive function overloading.

\paragraph{Template Instantiation Mapping} Template function calls are mapped to their instantiated forms when possible, though complete template resolution may require full compilation context.

\begin{algorithm}
\caption{Function Name Resolution}
\label{alg:name-resolution}
\begin{algorithmic}[1]
\Require Call site $s$, Available functions $F$, Context $ctx$
\Ensure Resolved function $f$ or $\text{null}$

\State $\text{candidates} \gets \text{FindCandidatesByName}(s.\text{name}, F)$

\If{$|\text{candidates}| = 1$}
    \Return $\text{candidates}[0]$ \Comment{Unique match}
\EndIf

\Comment{Apply namespace resolution}
\State $\text{candidates} \gets \text{FilterByNamespace}(\text{candidates}, ctx.\text{namespace})$

\If{$|\text{candidates}| = 1$}
    \Return $\text{candidates}[0]$
\EndIf

\Comment{Apply parameter-based resolution}
\State $\text{candidates} \gets \text{FilterByParameters}(\text{candidates}, s.\text{args})$

\If{$|\text{candidates}| = 1$}
    \Return $\text{candidates}[0]$
\ElsIf{$|\text{candidates}| = 0$}
    \Return $\text{null}$ \Comment{No valid resolution}
\Else
    \Return $\text{SelectBestMatch}(\text{candidates}, s)$ \Comment{Heuristic selection}
\EndIf
\end{algorithmic}
\end{algorithm}

\section{Cycle Detection Algorithms}

Cycle detection in call graphs is essential for identifying recursive function calls and circular dependencies that may impact program behavior or analysis accuracy.

\subsection{Strongly Connected Components}

The system employs Tarjan's algorithm for detecting strongly connected components (SCCs), which efficiently identifies all cycles in the call graph:

\begin{theorem}[SCC Properties]
In a call graph $G$, every strongly connected component represents a set of mutually recursive functions. If $|SCC| > 1$, then there exists at least one cycle within the component.
\end{theorem}

\begin{algorithm}
\caption{Tarjan's Algorithm for SCC Detection}
\label{alg:tarjan-scc}
\begin{algorithmic}[1]
\Require Call graph $G = (V, E)$
\Ensure Set of strongly connected components $\mathcal{S}$

\State $\text{index} \gets 0$, $\text{stack} \gets \text{empty}$, $\mathcal{S} \gets \emptyset$
\State Initialize arrays: $\text{indices}[|V|]$, $\text{lowlinks}[|V|]$, $\text{onStack}[|V|]$

\For{each vertex $v \in V$}
    \If{$\text{indices}[v]$ is undefined}
        \State $\text{StrongConnect}(v)$
    \EndIf
\EndFor

\Function{StrongConnect}{$v$}
    \State $\text{indices}[v] \gets \text{index}$
    \State $\text{lowlinks}[v] \gets \text{index}$
    \State $\text{index} \gets \text{index} + 1$
    \State $\text{stack.push}(v)$, $\text{onStack}[v] \gets \text{true}$
    
    \For{each successor $w$ of $v$}
        \If{$\text{indices}[w]$ is undefined}
            \State $\text{StrongConnect}(w)$
            \State $\text{lowlinks}[v] \gets \min(\text{lowlinks}[v], \text{lowlinks}[w])$
        \ElsIf{$\text{onStack}[w] = \text{true}$}
            \State $\text{lowlinks}[v] \gets \min(\text{lowlinks}[v], \text{indices}[w])$
        \EndIf
    \EndFor
    
    \If{$\text{lowlinks}[v] = \text{indices}[v]$}
        \State $\text{component} \gets \emptyset$
        \Repeat
            \State $w \gets \text{stack.pop}()$
            \State $\text{onStack}[w] \gets \text{false}$
            \State $\text{component} \gets \text{component} \cup \{w\}$
        \Until{$w = v$}
        \State $\mathcal{S} \gets \mathcal{S} \cup \{\text{component}\}$
    \EndIf
\EndFunction

\Return $\mathcal{S}$
\end{algorithmic}
\end{algorithm}

\subsection{Cycle Classification}

The system classifies detected cycles into several categories based on their characteristics:

\paragraph{Direct Recursion} Functions that call themselves directly, representing the simplest form of recursive behavior.

\paragraph{Mutual Recursion} Cycles involving two functions that call each other, common in algorithms like mutual recursive descent parsers.

\paragraph{Complex Cycles} Cycles involving three or more functions, which may indicate complex algorithmic patterns or potential design issues.

\paragraph{Conditional Cycles} Cycles that may not always execute due to conditional logic, identified through control flow analysis when available.

\begin{figure}[htbp]
\centering
\begin{tikzpicture}[
    node distance=2cm,
    function/.style={circle, draw, text centered, minimum width=1.2cm},
    call/.style={->,>=stealth,thick},
    cycle/.style={->,>=stealth,thick,red}
]
    % Direct recursion example
    \node[function] (rec) at (0,4) {recursive};
    \draw[cycle] (rec) to [loop above] (rec);
    
    % Mutual recursion example
    \node[function] (a) at (3,4) {A};
    \node[function] (b) at (5,4) {B};
    \draw[cycle] (a) to [bend left] (b);
    \draw[cycle] (b) to [bend left] (a);
    
    % Complex cycle example
    \node[function] (x) at (8,5) {X};
    \node[function] (y) at (10,4) {Y};
    \node[function] (z) at (8,3) {Z};
    \draw[cycle] (x) -> (y);
    \draw[cycle] (y) -> (z);
    \draw[cycle] (z) -> (x);
    
    % Labels
    \node at (0,2) {Direct Recursion};
    \node at (4,2) {Mutual Recursion};
    \node at (9,1.5) {Complex Cycle};
\end{tikzpicture}
\caption{Types of Cycles in Call Graphs}
\label{fig:cycle-types}
\end{figure}

\section{Tree Generation and Visualization}

The system generates tree-structured representations of call graphs for visualization and analysis purposes, handling the conversion from general graphs to tree structures.

\subsection{Tree Extraction from Graphs}

Since call graphs are generally directed acyclic graphs (DAGs) or may contain cycles, extracting meaningful tree representations requires sophisticated algorithms:

\begin{definition}[Call Tree]
A call tree $T = (N, E_T, r)$ derived from call graph $G$ is a rooted tree where:
\begin{itemize}
\item $N \subseteq V$ represents a subset of functions from the original graph
\item $E_T \subseteq E$ represents tree edges that form an acyclic structure
\item $r \in N$ is the designated root function
\end{itemize}
\end{definition}

\begin{algorithm}
\caption{Tree Extraction with Depth Limitation}
\label{alg:tree-extraction}
\begin{algorithmic}[1]
\Require Call graph $G = (V, E)$, Root function $r$, Maximum depth $d_{max}$
\Ensure Call tree $T = (N, E_T, r)$

\State $N \gets \{r\}$, $E_T \gets \emptyset$
\State $\text{queue} \gets [(r, 0)]$ \Comment{BFS queue with depth tracking}
\State $\text{visited} \gets \{r\}$ \Comment{Prevent cycles in tree}

\While{$\text{queue} \neq \emptyset$}
    \State $(v, \text{depth}) \gets \text{queue.dequeue}()$
    
    \If{$\text{depth} \geq d_{max}$}
        \textbf{continue} \Comment{Respect depth limit}
    \EndIf
    
    \For{each $(v, u) \in E$} \Comment{For each function called by $v$}
        \If{$u \notin \text{visited}$}
            \State $N \gets N \cup \{u\}$
            \State $E_T \gets E_T \cup \{(v, u)\}$
            \State $\text{visited} \gets \text{visited} \cup \{u\}$
            \State $\text{queue.enqueue}((u, \text{depth} + 1))$
        \EndIf
    \EndFor
\EndWhile

\Return $T = (N, E_T, r)$
\end{algorithmic}
\end{algorithm}

\subsection{Multi-Root Tree Generation}

For comprehensive analysis, the system can generate forests of call trees from multiple entry points:

\paragraph{Entry Point Identification} The system identifies potential entry points including main functions, exported functions, and callback functions.

\paragraph{Coverage Analysis} Multiple trees are generated to ensure comprehensive coverage of the codebase, with overlap analysis to identify shared function usage patterns.

\paragraph{Tree Merging} When appropriate, separate trees can be merged to show relationships between different subsystems.

\section{Complexity Analysis}

\subsection{Algorithmic Complexity}

The analysis algorithms exhibit different complexity characteristics based on graph structure and analysis parameters:

\begin{theorem}[Call Graph Construction Complexity]
Given $n$ functions and $m$ call relationships, the call graph construction algorithm has:
\begin{itemize}
\item Time complexity: $\bigO{n + m \cdot \log n}$ with efficient name resolution
\item Space complexity: $\bigO{n + m}$ for graph storage
\item Name resolution overhead: $\bigO{k \cdot \log n}$ per call, where $k$ is the average number of overloads
\end{itemize}
\end{theorem}

\begin{theorem}[Cycle Detection Complexity]
Tarjan's SCC algorithm for cycle detection has:
\begin{itemize}
\item Time complexity: $\bigO{n + m}$ where $n = |V|$ and $m = |E|$
\item Space complexity: $\bigO{n}$ for auxiliary data structures
\item Output complexity: $\bigO{n + c}$ where $c$ is the number of detected cycles
\end{itemize}
\end{theorem}

\subsection{Performance Optimization Strategies}

The system implements several optimization strategies to handle large-scale codebases efficiently:

\paragraph{Incremental Graph Construction} When analyzing modified codebases, the system can incrementally update call graphs rather than reconstructing them entirely.

\paragraph{Parallel Processing} Graph construction and cycle detection can be parallelized for large codebases by partitioning the analysis across multiple threads.

\paragraph{Caching and Memoization} Expensive operations such as name resolution and template instantiation analysis are cached to avoid redundant computation.

\paragraph{Lazy Evaluation} Tree generation and detailed analysis are performed on-demand rather than precomputing all possible trees.

\section{Statistical Analysis}

\subsection{Graph Metrics}

The system computes various metrics to characterize call graph properties:

\begin{definition}[Call Graph Metrics]
For a call graph $G = (V, E)$, define:
\begin{itemize}
\item \textbf{Density}: $\rho(G) = \frac{|E|}{|V|(|V|-1)}$ measures edge density
\item \textbf{Average degree}: $\bar{d}(G) = \frac{2|E|}{|V|}$ measures connectivity
\item \textbf{Clustering coefficient}: $C(G)$ measures local connectivity patterns
\item \textbf{Diameter}: $\text{diam}(G)$ measures maximum shortest path length
\end{itemize}
\end{definition}

\paragraph{Centrality Measures} The system computes various centrality measures to identify important functions:
\begin{itemize}
\item \textbf{In-degree centrality}: Functions called by many others
\item \textbf{Out-degree centrality}: Functions that call many others
\item \textbf{Betweenness centrality}: Functions that lie on many shortest paths
\item \textbf{PageRank centrality}: Functions with high recursive importance
\end{itemize}

\subsection{Distribution Analysis}

Statistical analysis of call patterns provides insights into codebase characteristics:

\paragraph{Degree Distribution} Analysis of in-degree and out-degree distributions reveals whether the codebase exhibits scale-free or other characteristic network properties.

\paragraph{Path Length Distribution} Distribution of shortest path lengths between functions indicates the hierarchical depth and modularity of the codebase.

\paragraph{Component Size Distribution} Analysis of strongly connected component sizes reveals the extent of recursive patterns in the codebase.

\begin{figure}[htbp]
\centering
\begin{tikzpicture}
    \begin{axis}[
        width=10cm,
        height=6cm,
        xlabel={Function In-Degree},
        ylabel={Frequency},
        title={Example Degree Distribution Analysis},
        grid=major,
        legend pos=north east
    ]
        \addplot[blue,thick] coordinates {
            (0,150) (1,89) (2,45) (3,23) (4,12) (5,8) (6,4) (7,2) (8,1)
        };
        \addlegendentry{Observed Distribution}
        
        \addplot[red,dashed,thick] coordinates {
            (0,145) (1,85) (2,50) (3,30) (4,18) (5,11) (6,7) (7,4) (8,2)
        };
        \addlegendentry{Power Law Fit}
    \end{axis}
\end{tikzpicture}
\caption{Statistical Analysis of Call Graph Properties}
\label{fig:degree-distribution}
\end{figure}

These analysis algorithms provide the foundation for comprehensive understanding of C++ codebase structure and enable sophisticated validation and reporting capabilities described in subsequent chapters.